Question

Assertion (A) : If probability of happening of an event is 0.2p, p $\gt $ 0, then p can't be more than 5.
Reason (R) : P(\(\overline{E}\)) = 1 – P(E) for an event E.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

To easily check if the Reason is the correct explanation for the Assertion, read the Assertion statement, insert the word "BECAUSE", and then read the Reason statement:
"If the probability of an event is \(0.2p\), then \(p\) cannot be more than 5 BECAUSE the probability of a complementary event is \(1 - P(E)\)."
This combined statement does not make logical sense. This confirms that the Reason is not the correct explanation of the Assertion!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The topic of this question is Probability.
This is an Assertion-Reason style question where we need to independently evaluate the mathematical truth of two statements: the Assertion (A) and the Reason (R).
If both statements are true, we must then determine whether the Reason (R) provides the correct logical explanation for the Assertion (A).

Step 2: Key Formula or Approach:
1. The probability of any event \(E\), denoted by \(P(E)\), is bounded. It must lie between 0 and 1 inclusive:
\[ 0 \le P(E) \le 1 \] 2. The sum of the probability of an event happening and the probability of it not happening (the complementary event) is always equal to 1:
\[ P(E) + P(\overline{E}) = 1 \implies P(\overline{E}) = 1 - P(E) \]

Step 3: Detailed Explanation:

• Let us analyze Assertion (A):
We are given that the probability of an event happening is \(P(E) = 0.2p\), where \(p \gt 0\).
Since the probability of any event cannot exceed 1, we must have:
\[ P(E) \le 1 \] Substitute the expression for \(P(E)\):
\[ 0.2p \le 1 \] \[ \frac{2}{10}p \le 1 \] \[ 2p \le 10 \] \[ p \le 5 \] This shows that \(p\) cannot be greater than 5. Thus, Assertion (A) is mathematically true.

• Let us analyze Reason (R):
The statement claims that \(P(\overline{E}) = 1 - P(E)\) for any event \(E\).
This is a standard definition and a fundamental formula for complementary events in probability. Thus, Reason (R) is mathematically true.

• Check if Reason (R) is the correct explanation for Assertion (A):
To prove that \(p \le 5\) in the Assertion, we used the boundary property that the maximum value of any probability is 1 (\(P(E) \le 1\)).
We did not use the formula for complementary events (\(P(\overline{E}) = 1 - P(E)\)) to derive or explain why \(p\) cannot exceed 5.
Therefore, while both statements are true, the Reason is not the correct explanation of the Assertion.


Step 4: Final Answer:
Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
Therefore, the correct option is (B).